C/D
y
1
[
n
]
x
1
(
t
)
x
2
(
t
)
T
=12kHz
C/D
C
H
(
z
)
x
1
(
t
)
x
2
(
t
)
T
1
=12KHz
C/D
A
T
2
=6KHz
B
y
2
[
n
]
2
2
x
[
n
]
y
1
[
n
]
H
1
(
e
j
ω
)
H
2
(
e
j
ω
)
x
[
n
]
Homework Set #3: Continuous to Discrete Time Translation
1.
The signal
x
c
(
t
) = 2cos(200
π
t
) is sampled with sampling periods,
(a)
T
= 1/300;
(b)
T
= 1/150;
(c)
T
= 1/100;
(d)
T
= 1/50 seconds, what are the resulting sampling sequences,
x
[
n
]?
2.
The signal
x
c
(
t
) = sin(200
π
t
) + cos(50
π
t
) is sampled with sampling periods,
(a)
T
= 1/500;
(b)
T
= 1/200;
(c)
T
= 1/100;
(d)
T
= 1/50 seconds, what are the resulting sampling sequences,
x
[
n
]?
3.
Consider the system show below. For each of the following input signal
x
[
n
], indicate whether
the output
x
r
[
n
] =
x
[
n
]:
(a)
x
[
n
] = cos(
π
n
/5);
(b)
x
[
n
] = cos(
π
n
/4);
(c)
x
[
n
] = cos(
π
n
/3);
(d)
x
[
n
] =
n
cos(
π
n
/2);
(e)
x
[
n
] = sin(
π
n
)
4.
Consider audio signal sampling and reconstruction, we plan to design 6time oversampling
system (by using ideal digital low pass filter) to reduce the burden of the design of analog filters.
The original signal sampling and reconstruction systems are shown as:
(a) Specify the original frequency responses of
H
a
(
j
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 Lowpass filter, Analog Filters, signal xc

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